Transcript Renormalization-group Method Applied to Derivation and

Application of the Renormalizationgroup Method for the Reduction of
Transport Equations
Teiji Kunihiro(YITP, Kyoto)
Renormalization Group 2005
Aug. 29 – Sep. 3, 2005
Helsinki, Finland
Based on:
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T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179
T.K.,Jpn. J. Ind. Appl. Math. 14 (’97), 51
T.K.,Phys. Rev. D57 (’98),R2035
T.K. and J. Matsukidaira, Phys. Rev. E57 (’98),
4817
• S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000),
236
• Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24
Contents
• Introduction; merits of RG
• RG equation v.s. envelope eq.
• A simple example for RG resummation
and derivation of the slow (amplitude and
phase) dynamics
• A generic example
• Fluid dynamic limit of Boltzmann eq.
• Brief summary and concluding remarks
The RG/flow equation
The yet unknown function
 , which
scale.
g
is solved exactly and inserted into
then becomes valid in a global domain of the energy
The merits of the Renormalization Group/Flow eq:
even for evolution equations appearing other fields!
The purpose of the talk:
(1) Show that the RG gives a powerful and systematic
method for the reduction of dynamics;
useful for construction of the attractive slow
manifold.
(2) Apply the method to reduce the fluid dynamics from
the Boltzmann equation.
(3) An emphasis put on the relation to the classical
theory of envelopes；
the resummed solution obtained through the RG is
the envelope of the set of solutions given in the
perturbation theory.
Geometrical Image of the Reduction of Dynamics
dim M  m  n
invariant
(attractive)
manifold
Y.Kuramoto(’89)
c.f. N.N.Bogoliubov
(a) Notion of inv. manifold
(b) Derivation of Boltzmann
equation from the Liouville
equation.
(c) Fluid dyn. from Boltzmann
A geometrical interpretation:
T.K. (’95)
construction of the envelope of the perturbative solutions
C : F ( x, y, , C( ))  0?
E: The envelope of C
G=0
y0
F ( 0 )  0 F ( 0 ')  0
x0
x
The envelop equation: dF / d 0  0
RG eq.
the solution is inserted to F with the condition
 0  x0
G( x, y)  F ( x, y, C( x))
the tangent point
A simple example:resummation
and extracting slowdynamics
T.K. (’95)
the dumped oscillator!
a secular term appears, invalidating P.T.
Secular terms appear again!
With I.C.:
; parameterized by the functions,
A(t0 ), (t0 )  t0   (t0 )
Let us try to construct the envelope function
terms divergent
invalidate functions,
the pert. theroy,
ofThe
thesecular
set of locally
like the log-divergence
Parameterized
by t0 ! in QFT!
:
The envelop function
an approximate but
global solution in contrast to the pertubative solutions
which have secular terms and valid only in
local domains.
c.f. Chen et al (’95)
Notice also the resummed nature!
More generic example
S.Ei, K. Fujii & T.K.(’00)
Def.
P the projection onto the kernel
P  Q 1
ker
A
Parameterized with m variables,
Instead of n !
The would-be rapidly changing terms can be eliminated by the
choice;
Then, the secular term appears only the P space;
a deformation of
the manifold.
Deformed (invariant) slow manifold:
A set of locally divergent functions parameterized by
t0 !
The RG/E equation u / t0 t t  0 gives the envelope, which is
0
globally valid:
The global solution (the invariant manifod):
We have derived the invariant manifold and the slow dynamics
on the manifold by the RG method.
Extension; (a) A Is not semi-simple. (2) Higher orders. (Ei,Fujii and T.K.
Ann.Phys.(’00))
Layered pulse dynamics for TDGL and NLS.
The fluid dynamics limit of
the Boltzmann equation
Liouville equation
Boltzman equation
Slower dynamics
The basics of Boltzmann equation:
the coll. Integral:
the symmetry of the cross section:
Hydro dyn.
The conservation laws:
the particle number
the momentum
The collision invariant:
the kin. energy
Notice; this is
only formal,
because the
distribution
function is not
solved!
H-function and the equilibrium
If ln f is collision invariant, the entropy (-H) does
not change.
This is the case when f (r, v, t ) is a local equilibrium
Function.
The reduction of Boltzmann eq. to
Fluid dynamical equation
T.K. (’99);Y.Hatta and T.K.(’02)
Suppose that the system is an old system and the
Space-time dependence of the distribution function
is now slow.
I.C.
Pert. Exp.
The 0-th order:
We choose the stationary solution:
Local Maxwellian!
The first order eq.:
Def. of the lin.op.A:
Def. the inn. prod.
P :the projection onto Ker A.
Q  1 P
The 1st order solution:
the secular term
Deformation from the
local equilibrium dist.
Applying the RG/E equation,
This is the master equation giving the time evolution of
which constitute the fluid dynamic equation!
In fact, taking the inner product with the elements
of Ker A, i.e.,
,
with
Euler Eq.
The higher order:
Navier-Stokes equation with a dissipation.
Y.Hatta and T.K. Ann. Phys.298,24 (2002)
Interesting to apply to derive the relativistic
Fluid dynamics with dissipations.
Brief Summary and concluding remarks
(1)The RG v.s. the envelop equation
(2) The RG eq. gives the reduction of dynamics and
the invariant manifold.
(3) The RG eq. was applied to reduce the Boltzmann
eq. to the fluid dynamics in the limit of a small
space variation.
Other applications:
a. the elimination of the rapid variable from FockerPlanck eq.
b. Derivation of Boltzmann eq. from Liouvill eq.
c. Derivation of the slow dynamics around bifurcations
and so on. See for the details,Y.Hatta and T.K.,
Ann. Phys. 298(2002),24
Some references
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T.K. Prog. Theor. Phys. 94 (’95), 503; 95(’97), 179
T.K.,Jpn. J. Ind. Appl. Math. 14 (’97), 51
T.K.,Phys. Rev. D57 (’98),R2035
T.K. and J. Matsukidaira, Phys. Rev. E57 (’98), 4817
S.-I. Ei, K. Fujii and T.K., Ann. Phys. 280 (2000), 236
Y. Hatta and T. Kunihiro, Ann. Phys. 298 (2002), 24
L.Y.Chen, N. Goldenfeld and Y.Oono,
PRL.72(’95),376; Phys. Rev. E54 (’96),376.